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Exploring errors in college mathematics courses.

Article Excerpt
Abstract

In a mathematics course for prospective elementary teachers, one university mathematics professor discussed his students' errors in a way that valued yet extended their mathematical thinking. He used errors to help students reconceptualize mathematics problems, explore contradictions, and pursue alternative strategies.

Introduction

Deciding how to deal with student errors is often difficult. Many teachers worry that discussing errors will make the student who made the error feel embarrassed, while others argue that discussing errors could introduce confusion into an otherwise cohesive discussion. This paper provides examples of how one university instructor navigated a discussion of errors in a way that supported his students' growth in mathematical understanding while preserving the value in his students' reasoning.

One team of researchers found that exploring errors in particular ways (see a, b, and c, below) was one of four features of mathematical discourse that made a difference in students' growth in understanding of fractious (Kazemi & Stipek, 2001, Stipek et al., 1998). In classrooms with the greatest growth in students' understanding, teachers used errors as a way for students to: a) reconceptualize problems; b) explore contradictions; and c) pursue alternative strategies.

For example, a fifth grade teacher created opportunities for her students to explore contradictions by verifying whether solutions were correct (Kazemi & Stipek, 2001). She responded to a student's mistake, which was thinking that 6/8 and 1 1/8 were both correct solutions to a problem, by encouraging all students to explain whether and why 6/8 and 1 1/8 were or were not equivalent. By encouraging the students in the class to reason about the two solutions, the teacher provided an opportunity for the students to engage in mathematical inquiry. Mistakes were used as opportunities for students to clarify and extend their thinking.

In contrast, Kazemi & Stipek (2001) found that in the classrooms with lesser growth in student understanding, errors were either ignored until a satisfactory solution was offered, or the teacher rather than the students provided the reasons that strategies were mathematically incorrect. In so doing, these teachers reduced the opportunities for students to reason mathematically. In high- growth classrooms, mistakes were not only viewed as a normal part of learning, but teachers pressed students to critically analyze both appropriate and flawed student reasoning.

This paper serves to contribute to the knowledge base about discourse, and in particular, the discussion of errors, by focusing on a university mathematics professor's class; most research on discourse has focused on K-12 classrooms (e.g., Nathan & Knuth, 2003; McClain & Cobb, 2001, Sherin et al., 2000, Kazemi, 1998, Yackel & Cobb, 1996). Although there are principles about discourse that university mathematics professors can garner by reading about...

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